In this dissertation, we worked on extending time series outlier detection methodology to spatial data. An integral part of the outlier detection algorithm is a hypothesis test for the presence of at least one outlier in the data. The distribution of the corresponding test statistic is not known and as a result the critical value corresponding to a size α test is estimated by approximating the tail probability for the test statistic. We identified and studied two methods of approximating the tail probability for the test statistic in the case when the parameters of the underlying spatial process are known. These approximations are based on bounds on the tail probability of the maxima of a discretely sampled Gaussian random field. We also study the distribution of the test statistic in the case when the parameters of the underlying spatial process are unknown and are estimated using maximum likelihood.